Acta Polytechnica Vol. 43 No. l/2003

A Short Note on Non-isothermal
Diffusion Models
T. Ficker

Asymptotic behaviour of the DIAL and DMLnon-isothermal mod,els, deriaed preaiously for the dffision of uater aapour through a porous
buil.ding structure, is studied under the assumption that the i,nitially non-isothennal structure becomes purely isothermal.

Keyuords: Isothermal and non-isothermal difiuions, dffision models, condensation in building structures.

I Introduction
In a previous research communication [], the DIAL and

DRAL non-isothermal difiLsion models were derived and
compared with the standard isothermal models commonly
used in building thermal technology. In a further related
communication [2] we applied these models to the Glaser
condensation scheme. The Glaser scheme enables, among
others, an assessment of condensate for a one-year period.
Naturally, throughout the year the structure is mostly exposed
to non-isothermal states but, especially in the summer season,
it may be subjected to a purely isothermal state. The DIAL
and DRAL models provide the basic relations solely for
non-isothermal conditions, i.e., they contain different tem-
peratures ?,, T, belonging to the opposite sides of the
structure. When approaching the isothermal state (?r-+ T,),
the relations give an uncertain expression 0/0 and, at first
sight, it is not clear how these relations should be applied to
the isothermal state (Zr= Tf T).

This short communication is aimed at deriving asympto-
tic DIAL and DRAL relations holding for the isothermal state
of a building structure possessing the diffusion resistance
factor p.

2 Asymptotic DIAL relations
For the DIAL model the generalised diffusion resistance

$ and diffusion'conductivity'Djip read [l]

3 Asymptotic DRAL relations
Repeating the same procedure as in the foregoing section,

we can rewrite the original DRAL non-isothermal relations []
n"n =J-, D"o =@ ,'' -", (4)-'eII 

D.ff ' 
"ttt 

l, T,)-' - Trl-'

using the 
',Hospital 

rule 
d (7, _Tr\

ti- ,l -? = li^ ,dr2' " =
rr+r,T|-n -Tt-n ,-r, 

h(d-, -4t-,; (5)

h*
into the purely isothermal relations (Tr=Tt=71

d h -, * -t.StReff =*-, Deff=-t =-I .uefl P P
Relation (6) exactly corresponds to the

(6)

isothermal

-* d ^* (2 -n)kpo Tt-Tz'-\eff=: ' ueff =-Deff tLR" rf- -r;-'
Fa =980665Pa,k =8.9?18'lol0 *2s-1K-t 

81,, 
= l8l.

To determine the relations describing the isothermal state
(Tr=Tt:71it is necessary to use the I'Hospital rule

@ /^ .T\

^ lt1 - t21dlo,. Tt -Tqll[I='-=-
Tz-+rr 7rz-n -rz-n '"'#('?-" -rr'*)

ti^ *' z
-fT A /^9-n -9-n\

dr, \'t
(9\

=Tt'=rn-tZ-"- 2-"'
which leads to the following result

4o =+, niu = k* rn; - k F: ro'tr.Dir PR" PR"
62

IMjIDR result [l], which confirms the consistency of the de-
veloped models.

4 Conclusion
Derived asymptotic relations (3) and (6) rePresent a neces-

sary complement for the DiAL and DRAL non-isotherrut'l
models when they are faced with the task of estimating the
water condensate inside building structures exposed to purely
isothermal conditions. Such problems may appear in building
thermal technology when calculating the one-year balance of
condensate inside the building envelopes.

References
tll Ficker, T., Pode5vov6, Z.: Modek for Non-lsothermal

Steady-Stnte Difusion in Porous Build'ing Materials. Acta
Polytechnica (accepted for publication).

t2l Ficker, T., Pode5vov6, Z.: Modified Glaser's Condensation
Moful. Acta Polytechnica (accepted for publication).

Assoc. Prof. RNDr. Tomd5 Ficker, DrSc.
phone: + 420 541 147 661
e-mail: fyfi c@fce.vutbr.cz

Department of Physics
University of Technology
Faculty of Civil Engineering
Zizkova 17
662 37 Brno, Czech Republic

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